3. The attempt at a solution
I got ρ = 0.1782 (approx) . I've been struggling a lot with this problem even though I don't have much work here to show for it. Most of the couple hours (literally hours) I've spent staring at this problem has been spent scouring the book for various equations that look like they may possibly help. I just can't figure out how to get the velocity (v) or the k without being given one of the two. Also I apologize for having to type in pi. I couldn't find the symbol in the available options and I don't see it on my keyboard. I also did some simple conversions from mm to meters, and so on.

There is a symbol for pi (π) near the middle of the first line of the available symbol options.

I see the π now. Thank you. I was looking for a squiggly line on the top; that's why I missed it. Thanks for the response altogether really. I saw that equation before. I just didn't remember/bother to write it down because the one definition of the adiabatic constant (idk which of your symbols you use for that, because none of them look that similar to the page you linked me. maybe Ψ or ϒ?) that I could find was γ = β / P(initial / equilibrium), and the definition for β i'm using is β = v2ρ. It's a circle. I read your link fully btw. Please, more help

I took another gander at the equation for the velocity of sound through gas that you linked again. I noticed an alternate definition for the bulk mod β. Can you please confirm whether the following is true?
β = P(equilibrium) / ρ
γ = β / P(equilibrium)I used these two definitions in a problem earlier, but I can't shake the feeling at least one of them must be wrong, because wouldnt the P(equilibrium) cancel in the definition for the adiabatic constant γ and it would equal nothing more than 1 / ρ ? ρ being density of course.

The constant ##\gamma## has a specific value for all monatomic gases, such as He. If you scroll down to about the middle of the link in post #2, you will see the value.

Your value for the density of He at the pressure and temperature given in the problem does not look correct if you are using units of kg/L. However, it looks correct for SI units. What would be the units for density in SI units?